Log Transform Deliverable
Dong Chen
10-12
1 Introduction
It is common in the analysis of biological data to log transform data representing concentrations or data representing dose response. Here we are going to perform logarithmic transformation and explore the problem.
2 Method
2.1 Part 1
For each distribution below, generate a figure of the PDF and CDF. Mark the mean and median in the figure.
For each distribution below, generate a figure of the PDF and CDF of the transformation Y = log(X) random variable. Mark the mean and median in the figure. You may use simulation or analytic methods in order find the PDF and CDF of the transformation.
For each of the distributions below, generate 1000 samples of size 100. For each sample, calculate the geometric and arithmetic mean. Generate a scatter plot of the geometic and arithmetic sample means. Add the line of identify as a reference line.
Generate a histogram of the difference between the arithmetic mean and the geometric mean.
2.1.1 Distribution 1
X ∼ GAMMA(shape = 3, scale = 1)
shape=3
scale=1
x=seq(0,10,by=0.1)
pdf=dgamma(x,shape,scale)
mean.gam1=shape*scale
median.gam1=qgamma(0.5,shape,scale)
plot(x,pdf,type = "l",main="PDF of Gamma Distribution")
#Mark the mean
abline(v=mean.gam1,col="red")
text(4, .05, "mean")
#Mark the median
abline(v=median.gam1,col="blue")
text(2, .1, "median")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked.
x=seq(0,15,by=0.1)
cdf=pgamma(x,shape,scale)
plot(x,cdf,type = "l",main="CDF of Gamma Distribution")
#Mark the mean
abline(v=mean.gam1,col="red")
text(4, .05, "mean")
#Mark the median
abline(v=median.gam1,col="blue")
text(2, .1, "median")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked.
data1=rgamma(10000,shape,scale)
#hist(data1,breaks=100)
hist(log(data1),breaks=100)
abline(v=mean(log(data1)),col="red")
abline(v=median(log(data1)),col="blue")
Figure of the PDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
plot(ecdf(log(data1)))
abline(v=mean(log(data1)),col="red")
abline(v=median(log(data1)),col="blue")
Figure of the CDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
art.mean=NA
geo.mean=NA
for (i in 1:1000){
data2=rgamma(100,shape,scale)
art.mean[i]=mean(data2)
geo.mean[i]=exp(mean(log(data2)))
}
plot(c(1:1000),art.mean)
plot(c(1:1000),geo.mean)
Scatter plot of the geometic and arithmetic sample means.
plot(geo.mean,art.mean)
abline(0,1)
Add a Reference line with geometic and arithmetic sample means.
hist(art.mean)
hist(geo.mean)
hist(art.mean-geo.mean)
Histogram of the difference between the arithmetic mean and the geometric mean.
2.1.2 Distribution 2
X ∼ LOG NORMAL(μ =  − 1, σ = 1)
lnmean=-1
lnsd=1
x=seq(0,10,by=0.01)
pdf=dlnorm(x,lnmean,lnsd)
mean.gam2=exp(lnmean+lnsd^2/2)
median.gam2=qlnorm(0.5,lnmean,lnsd)
plot(x,pdf,type = "l",main="PDF of Log Normal Distribution")
#Mark the mean
abline(v=mean.gam2,col="red")
#Mark the median
abline(v=median.gam2,col="blue")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked. Red line stands mean and blue line stands median.
x=seq(0,15,by=0.1)
cdf=plnorm(x,lnmean,lnsd)
plot(x,cdf,type = "l",main="CDF of Log Normal Distribution")
#Mark the mean
abline(v=mean.gam2,col="red")
#Mark the median
abline(v=median.gam2,col="blue")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked.
data3=rlnorm(10000,lnmean,lnsd)
#hist(data1,breaks=100)
hist(log(data3),breaks=100)
abline(v=mean(log(data3)),col="red")
abline(v=median(log(data3)),col="blue")
Figure of the PDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
plot(ecdf(log(data3)))
abline(v=mean(log(data3)),col="red")
abline(v=median(log(data3)),col="blue")
Figure of the CDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
art.mean=NA
geo.mean=NA
for (i in 1:1000){
data4=rlnorm(100,lnmean,lnsd)
art.mean[i]=mean(data4)
geo.mean[i]=exp(mean(log(data4)))
}
plot(c(1:1000),art.mean)
plot(c(1:1000),geo.mean)
Scatter plot of the geometic and arithmetic sample means.
plot(geo.mean,art.mean)
abline(0,1)
Add a Reference line with geometic and arithmetic sample means.
hist(art.mean)
hist(geo.mean)
hist(art.mean-geo.mean)
Histogram of the difference between the arithmetic mean and the geometric mean.
2.1.3 Distribution 3
X ∼ UNIFORM(0, 12)
min=0
max=12
x=seq(-5,15,by=0.01)
pdf=dunif(x,min,max)
mean.gam3=(min+max)/2
median.gam3=qunif(0.5,min,max)
plot(x,pdf,type = "l",main="PDF of Uniform Distribution")
#Mark the mean
abline(v=mean.gam3,col="red")
#Mark the median
abline(v=median.gam3,col="blue")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked. Red line stands mean and blue line stands median.
x=seq(-5,15,by=0.1)
cdf=punif(x,min,max)
plot(x,cdf,type = "l",main="CDF of Uniform Distribution")
#Mark the mean
abline(v=mean.gam3,col="red")
#Mark the median
abline(v=median.gam3,col="blue")
The Figure above is PDF of Gamma Distribution. Mean and median have been marked.
data5=runif(10000,min,max)
#hist(data1,breaks=100)
hist(log(data5),breaks=100)
abline(v=mean(log(data5)),col="red")
abline(v=median(log(data5)),col="blue")
Figure of the PDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
plot(ecdf(log(data5)))
abline(v=mean(log(data5)),col="red")
abline(v=median(log(data5)),col="blue")
Figure of the CDF of the transformation Y = log(X) random variable. Red line stands mean and blue line stands median.
art.mean=NA
geo.mean=NA
for (i in 1:1000){
data6=runif(100,min,max)
art.mean[i]=mean(data6)
geo.mean[i]=exp(mean(log(data6)))
}
plot(c(1:1000),art.mean)
plot(c(1:1000),geo.mean)
Scatter plot of the geometic and arithmetic sample means.
plot(geo.mean,art.mean)
abline(0,1)
Add a Reference line with geometic and arithmetic sample means.
hist(art.mean)
hist(geo.mean)
hist(art.mean-geo.mean)
Histogram of the difference between the arithmetic mean and the geometric mean.
2.2 Part 2
Show that if Xi > 0 for all i, then the arithmetic mean is greater than or equal to the geometric mean.
Hint: Start with the sample mean of the transformation Yi = log (Xi).
Method 1
Method 2
2.3 Part 3
What is the correct relationship between E [log (X)] and log(E[X])? Is one always larger? Equal? Explain your answer.
